Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,17)(16,23)(18,22)(19,25)(20,26)(21,27)(24,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,24,21)(18,22,26,20)(23,27,28,25);(6,9)(7,11)(13,16)(14,19)(15,21)(17,24)(18,27)(20,28)(22,23)(25,26);(5,7,8,11)(6,10,9,12)(14,15)(16,24)(20,22)(25,27);(6,9)(10,12)(14,15)(19,21)(20,22)(23,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420178);